The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. The basic Theil index TT is the same as redundancy in information theory which is the maximum possible entropy of the data minus the observed entropy. It is a special case of the generalized entropy index. It can be viewed as a measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility. It was proposed by econometrician Henri Theil at the Erasmus University Rotterdam.
For a population of N "agents" each with characteristic x, the situation may be represented by the list xi (i=1,...,N) where xi is the characteristic of agent i. For example, if the characteristic is income, then xi is the income of agent i. The Theil index is defined as:
where is the mean income:
Equivalently, if the situation is characterized by a discrete distribution function fk (k=0,...,W) where fk is the fraction of the population with income k and W = Nμ is the total income, then and the Theil index is: